Differentiation Rules: math test, College

Rate this post

Math tests College : Differentiation Rules — Zyro the alien explorer of Planète Maths

Suggested time: 45 minutes. Out of 20 points. Calculator allowed only when the problem says so.

Sommaire

1 Product and quotient rules / 4 pts

Differentiate. (a) \( f(x)=(3x-1)(x^2+4) \). (b) \( g(x)=\dfrac{2x+1}{x-3} \), for \( x\neq 3 \).

2 Chain rule / 3 pts

Differentiate. (a) \( y=(x^3+2)^6 \). (b) \( y=e^{-3x}\cos(2x) \).

3 A logarithmic quotient / 3 pts

Let \( y=\dfrac{\ln x}{x^2} \) for \( x\gt 0 \). (a) Find \( y' \). (b) Find the exact \( x \)-value where the tangent line is horizontal.

4 Implicit differentiation / 3 pts

The point \( (3,3) \) lies on the curve \( x^3+y^3=6xy \). Find \( \dfrac{dy}{dx} \) in terms of \( x \) and \( y \), then the equation of the tangent line at \( (3,3) \).

5 Logarithmic differentiation / 4 pts

(a) For \( x\gt 0 \), differentiate \( y=x^{3x} \) and compute \( y'(1) \). (b) For \( y=\dfrac{(x+1)^4\sqrt{x-2}}{(3x-1)^2} \) with \( x\gt 2 \), write \( \dfrac{y'}{y} \) and evaluate it at \( x=3 \).